Central Limit Theory for Martingales via Random Change of Time.
NORTH CAROLINA UNIV AT CHAPEL HILL CENTER FOR STOCHASTIC PRECESSES
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This paper contains an exposition of the by now rather complete central limit theory for discrete parameter martingales providing new and efficient proofs. The basic idea is to start by proving a central limit theorem under quite restrictive conditions that the summands tend uniformly to zero and that the sums of squares converge uniformly and then to obtain the most general results by random change of time and truncation. The emphasis is on the sums of squares or squared variation process, and Burkholders square function inequality plays a crucial role in the development. In particular, this approach leads to a very short and direct proof of tightness. In the proofs we make much use of a result which is believed to be new and which binds together convergence to zero of sums and of sums of conditional expectations. In the final section, the results are extended to several dimensions, to mixing convergence, and to convergence to mixtures of normal distributions. Author
- Statistics and Probability